where \({u}^{f}\)  and \({u}^{p}\)  are the velocities of the fluid and the particle. Further, we assume that the fluid velocity is linearly distributed in the z-direction inspired by the work of Ref.[1]}. It needs to be mentioned that the drag force in the present work is based on the condition of a low Reynolds number in the dense granular flow.
The lubrication force [2]} between particle \(i\)  and \(j\)  in the normal and tangential directions are
\(\begin{split}{f}_{ij}^{lubr,n}&=6\pi \eta _{f}R_{\text{eff}}^{2}\frac{{v}_{n}^{rel}}{\delta _{g}},\ \ \textrm {and}\\{f}_{ij}^{lubr,t}&=6\pi \eta _{f}R_{\text{eff}}^{2}{v}_{t}^{rel}\left[\frac{8}{15}\ln {\frac{R_{\text{eff}}}{\delta _{g}}+0.9588}\right],\\\end{split}\) 
